ICAP Summer School, Paris, Three lectures on quantum gases. Wolfgang Ketterle, MIT

Transcription

1 ICAP Summer School, Paris, 2012 Three lectures on quantum gases Wolfgang Ketterle, MIT

2 Cold fermions

3 Reference for most of this talk: W. Ketterle and M. W. Zwierlein: Making, probing and understanding ultracold Fermi gases. in Ultracold Fermi Gases, Proceedings of the International School of Physics "Enrico Fermi", Course CLXIV, Varenna, June 2006, edited by M. Inguscio, W. Ketterle, and C. Salomon (IOS Press, Amsterdam) 2008, pp ; e-print, arxiv: ; Rivista del Nuovo Cimento 31, (2008).

4 Li Na cooling movie Lithium Sodium

5 At absolute zero temperature Bosons Particles with an even number of protons, neutrons and electrons Bose-Einstein condensation atoms as waves superfluidity Fermions Particles with an odd number of protons, neutrons and electrons Fermi sea: Atoms are not coherent No superfluidity

6

7

8 Freezing out of collisions g (2) (r) Pair correlations in a Fermi gas: Distance λ db No interactions if range of potential is < λ db Elastic collisions suppressed below T pwave

9 fermion mixture s-wave identical fermions, p-wave Elastic cross section for K-40 (Jin group, PRL 1999)

10 Pairs of fermions Particles with an even number of protons, neutrons and electrons Two kinds of fermions Fermi sea: Atoms are not coherent No superfluidity

11 At absolute zero temperature Pairs of fermions Particles with an even number of protons, neutrons and electrons Bose-Einstein condensation atoms as waves superfluidity Two kinds of fermions Particles with an odd number of protons, neutrons and electrons Fermi sea: Atoms are not coherent No superfluidity

12 Weak attractive interactions Cooper pairs larger than interatomic distance momentum correlations BCS superfluidity Two kinds of fermions Particles with an odd number of protons, neutrons and electrons Fermi sea: Atoms are not coherent No superfluidity

13 E Free atoms Molecule Magnetic field Feshbach resonance

14 Disclaimer: Drawing is schematic and does not distinguish nuclear and electron spin. E Free atoms Molecule Magnetic field Feshbach resonance

15 Two atoms. E Free atoms Molecule Magnetic field Feshbach resonance

16 form a stable molecule E Free atoms Molecule Magnetic field Feshbach resonance

17 Atoms attract each other E Free atoms Molecule Magnetic field Feshbach resonance

18 Atoms repel each other Atoms attract each other E Free atoms Molecule Magnetic field Feshbach resonance

19 Atoms repel each other Atoms attract each other Force between atoms Scattering length Magnetic field Feshbach resonance

20 Feshbach loss and a (JPEG) Observation of a Feshbach resonance S. Inouye, M.R. Andrews, J. Stenger, H.-J. Miesner, D.M. Stamper-Kurn, WK, Nature 392 (1998).

21 Energy Atoms Molecules Magnetic field Atoms form stable molecules Atoms repel each other a>0 BEC of Molecules: Condensation of tightly bound fermion pairs Molecules are unstable Atoms attract each other a<0 BCS-limit: Condensation of long-range Cooper pairs

22 Atom pairs Electron pairs Bose Einstein condensate of molecules BCS Superconductor

23 ular BEC BCS superfluid

24 Magnetic field ular BEC BCS superfluid

25 ular BEC Crossover superfluid BCS superfluid

26 How do atoms pair?

27 Two-body bound states in 1D, 2D, and 3D 1D, 2D: bound state for arbitrarily small attractive well 3D: Well depth has be larger than threshold

28 Connection to the density of states In momentum space

29 Connection to the density of states In momentum space Short range potential: V(q)=V 0 for q<1/r Integrate over q, divide by common factor

30 Bound state for arbitrarily small V 0 only if integral diverges for E 0

31 Bound state for arbitrarily small V 0 only if integral diverges for E 0 In 2D (constant density of states): logarithmic divergence

32 The Cooper problem:

33 Two fermions with weak interactions on top of a filled Fermi sea Total momentum zero Total momentum non-zero 2q

34 Pauli blocking Compare with previous result for single particle bound state

35 Pauli blocking After replacing the bare interaction V 0 by the scattering length a

36 Cooper Pairing Consider two particles,, on top of a filled, inert Fermi sea Total momentum zero Total momentum non-zero Reduced density of states Much smaller binding energy The important pairs are those with zero momentum

37 BCS Wavefunction How can we find a state in which all fermions are paired in a self-consistent way? John Bardeen Leon N. Cooper John R. Schrieffer

38 BCS Wavefunction Many-body wavefunction for a condensate of Fermion Pairs: Spatial pair wavefunction Spin wavefunction Second quantization: Fourier transform: Pair wavefunction: Operators: Pair creation operator: Many-body wavefunction: a fermion pair condensate

39 is not a Bose condensate Commutation relations for pair creation/annihilation operators Occupation of momentum k pairs do not obey Bose commutation relations, unless BEC limit of tightly bound molecules

40 BCS Wavefunction Introduce coherent state / switch to grand-canonical description: and commute because Normalization: BCS wavefunction: with and

41 BCS Wavefunction and are identical!

42 Many-Body Hamiltonian Second quantized Hamiltonian for interacting fermions: Contact interaction: Fourier transform via BCS Approximation: Only include scattering between zero-momentum pairs Solve via 1) Variational Ansatz, 2) via Bogoliubov transformation

43 Variational Ansatz: Insert BCS wavefunction into Many-Body Hamiltonian. Minimize Free Energy: Result: E k Δ Gap equation:

44 Solution via Bogoliubov Transform BCS Hamiltonian is quartic: Introduce pairing field (mean field or decoupling approximation): small fluctuations (assumption) Neglect products (correlations) of those small fluctuations Define This plays the role of the condensate wavefunction

45 Solution via Bogoliubov Transform Rewrite Hamiltonian, drop terms quadratic in C s: Hamiltonian is now bilinear Solve via Bogoliubov transformation to quasiparticle operators: With the choice and we get Ground state energy Non-interacting gas of fermionic quasi-particles

46 Solution of the gap equation Looks similar to equation for bound state in 2D (and Cooper problem)

47 Gap equation: Solution of the gap equation Number equation: Simultaneously solve for µ and Δ

48 Solution of the gap equation BEC-side: Molecules BCS-side: Gap exponentially small

49 Critical temperature Can be derived from Bogoliubov Hamiltonian with fluctuations T C /T F Conventional SC: Superfluid 3 He: 10-3 High-T C SC: 10-2 High-T C Superfluid: 0.15 BEC-limit: BCS-limit:

50 Experimental realization of the BEC-BCS Crossover

51 Preparation of an interacting Fermi system in Lithium-6 Electronic spin: S = ½, Nuclear Spin: I = 1 (2I+1)(2S+1) = 6 hyperfine states Optical 1064 nm ν axial = Hz ν radial = Hz E trap = µk

52

53

54 BEC of Fermion Pairs (Molecules) T > T C T < T C T T C These days: Up to 10 million condensed molecules Boulder Nov 03 Innsbruck Nov 03, Jan 04 MIT Nov 03 Paris March 04 Rice, Duke M.W. Zwierlein, C. A. Stan, C. H. Schunck, S.M.F. Raupach, S. Gupta, Z. Hadzibabic, W. Ketterle, Phys. Rev. Lett. 91, (2003)

55 JILA, Nature 426, 537 (2003). MIT, PRL 91, (2003) Innsbruck, PRL 92, (2004). ENS, PRL 93, (2004).

56 Observation of Pair Condensates BEC-Side Resonance BCS-Side (above dissociation limit for molecules) Thermal + condensed pairs First observation: C.A. Regal et al., Phys. Rev. Lett. 92, (2004) M.W. Zwierlein, C.A. Stan, C.H. Schunck, S.M.F. Raupach, A.J. Kerman, W. Ketterle, Phys. Rev. Lett. 92, (2004).

57 Condensate Fraction vs Magnetic Field k F a > 1 M.W. Zwierlein, C.A. Stan, C.H. Schunck, S.M.F. Raupach, A.J. Kerman, W. Ketterle, Phys. Rev. Lett. 92, (2004).

58 How can we show that these gases are superfluid?

59 Vortex lattices in the BEC-BCS crossover Establishes superfluidity and phase coherence in gases of fermionic atom pairs B M.W. Zwierlein, J.R. Abo-Shaeer, A. Schirotzek, C.H. Schunck, W. Ketterle, Nature 435, (2005)

60 Superfluidity of fermions requires pairing of fermions Microscopic study of the pairs by RF spectroscopy

61 RF spectroscopy 3> >" hf 0 3> >" hf 0 +Δ >" >"

62 Dissociation spectrum measures the Fourier transform of the pair wavefunction Width (1/pair size) 2 Threshold (1/pair size) 2

63 Rf spectra in the crossover Standard superconductors ξ>> 1/k F High T c superconductors ξ 6 10 (1/k F ) Superfluid at unitarity ξ = 2.6 (1/k F ) Interparticle spacing ~ 3.1 (1/k F ) Molecular character of fermion pairs Confirms correlation between high T c and small pairs C. H. Schunck, Y. Shin, A. Schirotzek, W. Ketterle, Nature 454, 739 (2008).

64 Excitations in a superfluid Particles Holes

65 Δ2/2EF

66 How to inject quasi-particles near the Fermi surface? µ Population imbalance kt Δ -µ µ

67 RF Spectroscopy of a BCS superfluid Final state empty, measures integrated (over k) spectrum RF photon creates quasiparticle and free particle in third state quasiparticle dispersion free particle dispersion cold injected quasiparticles

68 Δ Δ 2 /2E F BCS limit: splitting becomes exactly Δ

69 Polarized Superfluid local double peak: Stokes and Anti-Stokes peak BCS T 0 limit: Splitting is exactly Δ Splitting allows a direct determination of the superfluid gap Δ Δ = 0.44 E F QMC (Carlson, Reddy 2008) Δ = 0.45 E F Related experiment JILA: RF photoemission A. Schirotzek, Y. Shin, C.H. Schunck, W.K., Phys. Rev. Lett. 101, (2008).

70 Now: Fermions with repulsive interactions

71 Feshbach Resonance Energy Atoms Molecules Magnetic field Atoms form stable molecules Atoms repel each other a>0 BEC Itinerant of Molecules: Ferromagnetism Condensation Stoner instability of tightly in a free bound gas fermion pairs Molecules are unstable Atoms attract each other a<0 BCS-limit: Condensation of long-range Cooper pairs

72 Itinerant Ferromagnetic Phase Transition in Ultracold gases Increasing k F a 2 2/3 Energy : Spin : Spin

73 Stoner model A Fermi gas with short-range repulsive interactions Kinetic energy

74 Mean-field approximation for interaction term:

75 local magnetization K.E. spin up K.E. spin down repulsive mean field interaction phase transition for

76 Prepared a two-component Fermi gas( ~ 0.65 million per each spin state) Vary repulsive interactions near Feshbach resonance located at 834 G

77 Three observations of non-monotonic behavior when approaching the Feshbach resonance Suggests that itinerant FM can occur for a free gas with short-range interactions First study of quantum magnetism in cold fermionic atoms Quantum simulation of a Hamiltonian for which even the existence of a phase transition is unknown BUT: Lifetime only 10 ms Molecular fraction 25 % Magnetic domains not resolved Ferromagnetic fluctuations vs. ferromagnetic ground state G.B. Jo, Y.R. Lee, J.H. Choi, C.A. Christensen, H. Kim, J. Thywissen, D.E. Pritchard, W.K., Science 325, (2009).

78 More recent work: No ferromagnetic transition Rapid decay into pairs Highly correlated gas, breakdown of mean field description Atoms with strong repulsion cannot be isolated from molecules C. Sanner, E.J. Su, W. Huang, A. Keshet, J. Gillen, W.K., Phys. Rev. Lett. 108, (2012)

79 Feshbach Resonance Energy Atoms Molecules Magnetic field Atoms form stable molecules Atoms repel each other a>0 BEC Itinerant of Molecules: Ferromagnetism Condensation Stoner instability of tightly in a free bound gas fermion pairs Molecules are unstable Atoms attract each other a<0 BCS-limit: Condensation of long-range Cooper pairs

80 Cold atomic gases provide the building blocks of quantum simulators Quantum engineering of interesting Hamiltonians Ultracold Bose gases: superfluidity (like 4 He) Ultracold Fermi gases (with strong interactions near the unitartiy limit): pairing and superfluidity (BCS, like superconductors) Optical lattices: crystalline materials Soon: magnetism in strongly correlated systems